Techniques of Integration

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Presentation transcript:

Techniques of Integration Chapter 9 Techniques of Integration Copyright © 2014, 2010, 2007 Pearson Education, Inc.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. Chapter Outline Integration by Substitution Integration by Parts Evaluation of Definite Integrals Approximation of Definite Integrals Some Applications of the Integral Improper Integrals Copyright © 2014, 2010, 2007 Pearson Education, Inc.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 9.2 Integration by Parts Copyright © 2014, 2010, 2007 Pearson Education, Inc.

Integration by Parts Using Integration by Parts Section Outline Integration by Parts Using Integration by Parts Copyright © 2014, 2010, 2007 Pearson Education, Inc.

Integration by Parts The following equation is the principle of integration by parts and is one of the most important techniques of integration. G(x) is an antiderivative of g(x). Copyright © 2014, 2010, 2007 Pearson Education, Inc.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. Using Integration by Parts EXAMPLE Evaluate. SOLUTION Our calculations can be set up as follows: Differentiate Integrate Then Copyright © 2014, 2010, 2007 Pearson Education, Inc.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. Using Integration by Parts CONTINUED Copyright © 2014, 2010, 2007 Pearson Education, Inc.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. Using Integration by Parts EXAMPLE Evaluate. SOLUTION Our calculations can be set up as follows: Then Notice that the resultant integral cannot yet be solved using conventional methods. Therefore, we will attempt to use integration by parts again. Copyright © 2014, 2010, 2007 Pearson Education, Inc.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. Using Integration by Parts CONTINUED Our calculations can be set up as follows: Then Therefore, we have Copyright © 2014, 2010, 2007 Pearson Education, Inc.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. Using Integration by Parts EXAMPLE Evaluate. SOLUTION Our calculations can be set up as follows: Then Copyright © 2014, 2010, 2007 Pearson Education, Inc.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. Using Integration by Parts CONTINUED Copyright © 2014, 2010, 2007 Pearson Education, Inc.